736 
MR. R. C. ROWE ON ABEL’S THEOREM. 
(iii) to find this value. 
The proof of (i) is most simply conducted by successively investigating the 
conditions 
(a) that the major term of any (say the ?’ th ) set should furnish a higher power of x 
(for the substitution y r ) than any other major term furnishes, 
(b) that this major term should furnish a higher power than any other not-major 
term furnishes. 
In investigating ( b ) the conditions of (a) are to be supposed to hold. It will only 
be found necessary to supply to them a slight additional restriction in order to satisfy (b). 
17. The condition for (a) is that whatever values (of course lying between 0 and 
n — 1 inclusive) are given to r and 5 we should have 
q Pr + pr<Tr > q Ps + p s Vr, 
where we have taken q Pr y Pr to be the major term of the r th set. 
We will write this, for brevity, in the form 
so 
that 
M+P w,-> [p J+ftw 
M=i f 
If we make successi vely the substitutions 
r=m+1 
s=m 
r—m 
s=m+ 1 
we find that the above inequality requires the following 
[/WiJ [pj ^ (p« pm+i)(r m+ i 
(pm P>ii+\)o"m 
If then we write 
we have 
[P'«+i] [pj—(p>« P»i+ i) r », 
r ,n or M+ ^ 
If we use also for p m —p ni+1 the abbreviation Bp m we have 
(A) 
CP"2+lJ l P»>\ — Spm-T,,, 
[pj — [Plj + ^Pl- T l+ • * • +§P/«-i*L«_i 
k=m— 1 
= [pi]+ 2 S pi.Tt . 
ic= 1 
(B) 
and it follows that 
